Séminaire Bourbaki Volume 2005/2006 Exposés 952-966

Most people, when they think of Nicolas Bourbaki, think first of his multi-volume "textbook", the Éléments de Mathématique. It may well be the case, however, that Bourbaki's most significant contribution to mathematics has been the long-running Séminaire that bears his name. Held each academic year in November, March, and June, the seminar is a venue for high-level exposition. Its yearly collection of "exposés" is a reliable source of information about what is going on in mathematics, at least from the point of view of the Bourbakistes.

The structure of the seminar is now standard: top mathematicians are invited to give talks, usually not about their own work, but about someone else's recent breakthrough. Important new theorems or concepts serve as the topics, so that the volumes of the seminar notes also serve as a sketch of the recent history of mathematics as it appeared each year.

The Séminaire has a complex publishing history, but for the last twenty or so years it has been published in the Société Mathématique de France's Asterisque series. The volume under review contains the notes from talks given in the 2005–06 academic year. The topics covered are quite varied, but show a slant towards traditional interests of Bourbaki: algebraic geometry and number theory are well represented, as are Lie theory and differential equations. But the perfect graph conjecture (here called the perfect graph theorem) gets an exposé, showing that the organizers are not restricted to those areas. The exposition is at a very high level, quite technical but often illuminating. Most of the articles are in French, of course.

This is an important series that most libraries should have.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.